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Let $R$ be an order in a number field. Two $R$-ideals $I$ and $J$ are weak equivalent if there exist (necessarily invertible) ideals $X$ and $Y$ such that $I X=J$ and $J Y=I$. This is equivalent to the localizations $I_{P}$ and $J_{P}$ being in the same ideal class of $R_{P}$ for every prime ideal $P$.

It is not difficult to deduce that weak equivalent ideals are then "finite" equivalent in the following sense: for every non-zero $\alpha \in R$, the finite $R$-modules $I/\alpha I$ and $J/\alpha J$ are isomorphic.

I would like to know if the converse is true.

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